// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_BICGSTAB_H
#define EIGEN_BICGSTAB_H

namespace Eigen {

namespace internal {

    /** \internal Low-level bi conjugate gradient stabilized algorithm
  * \param mat The matrix A
  * \param rhs The right hand side vector b
  * \param x On input and initial solution, on output the computed solution.
  * \param precond A preconditioner being able to efficiently solve for an
  *                approximation of Ax=b (regardless of b)
  * \param iters On input the max number of iteration, on output the number of performed iterations.
  * \param tol_error On input the tolerance error, on output an estimation of the relative error.
  * \return false in the case of numerical issue, for example a break down of BiCGSTAB. 
  */
    template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
    bool bicgstab(const MatrixType& mat, const Rhs& rhs, Dest& x, const Preconditioner& precond, Index& iters, typename Dest::RealScalar& tol_error)
    {
        using std::abs;
        using std::sqrt;
        typedef typename Dest::RealScalar RealScalar;
        typedef typename Dest::Scalar Scalar;
        typedef Matrix<Scalar, Dynamic, 1> VectorType;
        RealScalar tol = tol_error;
        Index maxIters = iters;

        Index n = mat.cols();
        VectorType r = rhs - mat * x;
        VectorType r0 = r;

        RealScalar r0_sqnorm = r0.squaredNorm();
        RealScalar rhs_sqnorm = rhs.squaredNorm();
        if (rhs_sqnorm == 0)
        {
            x.setZero();
            return true;
        }
        Scalar rho = 1;
        Scalar alpha = 1;
        Scalar w = 1;

        VectorType v = VectorType::Zero(n), p = VectorType::Zero(n);
        VectorType y(n), z(n);
        VectorType kt(n), ks(n);

        VectorType s(n), t(n);

        RealScalar tol2 = tol * tol * rhs_sqnorm;
        RealScalar eps2 = NumTraits<Scalar>::epsilon() * NumTraits<Scalar>::epsilon();
        Index i = 0;
        Index restarts = 0;

        while (r.squaredNorm() > tol2 && i < maxIters)
        {
            Scalar rho_old = rho;

            rho = r0.dot(r);
            if (abs(rho) < eps2 * r0_sqnorm)
            {
                // The new residual vector became too orthogonal to the arbitrarily chosen direction r0
                // Let's restart with a new r0:
                r = rhs - mat * x;
                r0 = r;
                rho = r0_sqnorm = r.squaredNorm();
                if (restarts++ == 0)
                    i = 0;
            }
            Scalar beta = (rho / rho_old) * (alpha / w);
            p = r + beta * (p - w * v);

            y = precond.solve(p);

            v.noalias() = mat * y;

            alpha = rho / r0.dot(v);
            s = r - alpha * v;

            z = precond.solve(s);
            t.noalias() = mat * z;

            RealScalar tmp = t.squaredNorm();
            if (tmp > RealScalar(0))
                w = t.dot(s) / tmp;
            else
                w = Scalar(0);
            x += alpha * y + w * z;
            r = s - w * t;
            ++i;
        }
        tol_error = sqrt(r.squaredNorm() / rhs_sqnorm);
        iters = i;
        return true;
    }

}  // namespace internal

template <typename _MatrixType, typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar>> class BiCGSTAB;

namespace internal {

    template <typename _MatrixType, typename _Preconditioner> struct traits<BiCGSTAB<_MatrixType, _Preconditioner>>
    {
        typedef _MatrixType MatrixType;
        typedef _Preconditioner Preconditioner;
    };

}  // namespace internal

/** \ingroup IterativeLinearSolvers_Module
  * \brief A bi conjugate gradient stabilized solver for sparse square problems
  *
  * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient
  * stabilized algorithm. The vectors x and b can be either dense or sparse.
  *
  * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
  *
  * \implsparsesolverconcept
  *
  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
  * and NumTraits<Scalar>::epsilon() for the tolerance.
  * 
  * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
  * 
  * \b Performance: when using sparse matrices, best performance is achied for a row-major sparse matrix format.
  * Moreover, in this case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
  * See \ref TopicMultiThreading for details.
  * 
  * This class can be used as the direct solver classes. Here is a typical usage example:
  * \include BiCGSTAB_simple.cpp
  * 
  * By default the iterations start with x=0 as an initial guess of the solution.
  * One can control the start using the solveWithGuess() method.
  * 
  * BiCGSTAB can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
  *
  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
  */
template <typename _MatrixType, typename _Preconditioner> class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType, _Preconditioner>>
{
    typedef IterativeSolverBase<BiCGSTAB> Base;
    using Base::m_error;
    using Base::m_info;
    using Base::m_isInitialized;
    using Base::m_iterations;
    using Base::matrix;

public:
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef _Preconditioner Preconditioner;

public:
    /** Default constructor. */
    BiCGSTAB() : Base() {}

    /** Initialize the solver with matrix \a A for further \c Ax=b solving.
    * 
    * This constructor is a shortcut for the default constructor followed
    * by a call to compute().
    * 
    * \warning this class stores a reference to the matrix A as well as some
    * precomputed values that depend on it. Therefore, if \a A is changed
    * this class becomes invalid. Call compute() to update it with the new
    * matrix A, or modify a copy of A.
    */
    template <typename MatrixDerived> explicit BiCGSTAB(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

    ~BiCGSTAB() {}

    /** \internal */
    template <typename Rhs, typename Dest> void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
    {
        m_iterations = Base::maxIterations();
        m_error = Base::m_tolerance;

        bool ret = internal::bicgstab(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error);

        m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
    }

protected:
};

}  // end namespace Eigen

#endif  // EIGEN_BICGSTAB_H
